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This text is devoted to the theory of varieties, which provides an important tool, based in universal algebra, for the classification of regular languages. In the introductory section, we present a number of examples that illustrate and motivate the fundamental concepts. We do this for the most part without proofs, and often without precise definitions, leaving these to the formal development of the theory that begins in Section 2. Our presentation of the theory draws heavily on the work of Gehrke, Grigorieff and Pin (2008) on the equational theory of lattices of regular languages. In the subsequent sections we consider in more detail aspects of varieties that were only briefly evoked in the introduction: Decidability, operations on languages, and characterizations in formal logic.
Let $Xsubset mathbb{P}^r$ be an integral and non-degenerate variety. Let $sigma _{a,b}(X)subseteq mathbb{P}^r$, $(a,b)in mathbb{N}^2$, be the join of $a$ copies of $X$ and $b$ copies of the tangential variety of $X$. Using the classical Alexander-Hir
Exponential varieties arise from exponential families in statistics. These real algebraic varieties have strong positivity and convexity properties, familiar from toric varieties and their moment maps. Among them are varieties of inverses of symmetri
This paper studies affine Deligne-Lusztig varieties in the affine flag manifold of a split group. Among other things, it proves emptiness for certain of these varieties, relates some of them to those for Levi subgroups, extends previous conjectures c
Motivated by the study of the secant variety of the Segre-Veronese variety we propose a general framework to analyze properties of the secant varieties of toric embeddings of affine spaces defined by simplicial complexes. We prove that every such sec
We study the anti-commuting variety which consists of pairs of anti-commuting $ntimes n$ matrices. We provide an explicit description of its irreducible components and their dimensions. The GIT quotient of the anti-commuting variety with respect to t